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Projectile motion practice problems

Projectiles: time of flight, maximum height and range from a launch speed, angle and height. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

2 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 9.671

medium

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 96 \) ft/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

x(t) = v₀cos θ · t and y(t) = 0 + v₀sin θ · t − ½gt².
Reviewed
\[ \left. \frac{d}{d t} \left(- 16 t^{2} + 48 t\right) \right|_{\substack{ t=\frac{3}{2} }} = 0 \]
The vertical velocity is 0 at t = 3/2.✓ Proved
\[ \left. - 16 t^{2} + 48 t \right|_{\substack{ t=\frac{3}{2} }} = 36 \]
The maximum height.✓ Proved
\[ \left. - 16 t^{2} + 48 t \right|_{\substack{ t=3 }} = 0 \]
It lands (y = 0) at t = 3.✓ Proved
\[ \left. 48 \sqrt{3} t \right|_{\substack{ t=3 }} = 144 \sqrt{3} \]
The horizontal distance then.✓ Proved

Answer \( H = 36 \approx 36.00,\ T = 3 \approx 3.000,\ \text{range} = 144 \sqrt{3} \approx 249.42 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.678

medium

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 20 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

x(t) = v₀cos θ · t and y(t) = 2 + v₀sin θ · t − ½gt².
Reviewed
\[ \left. \frac{d}{d t} \left(- \frac{49 t^{2}}{10} + 10 t + 2\right) \right|_{\substack{ t=\frac{50}{49} }} = 0 \]
The vertical velocity is 0 at t = 50/49.✓ Proved
\[ \left. - \frac{49 t^{2}}{10} + 10 t + 2 \right|_{\substack{ t=\frac{50}{49} }} = \frac{348}{49} \]
The maximum height.✓ Proved
\[ \left. - \frac{49 t^{2}}{10} + 10 t + 2 \right|_{\substack{ t=\frac{50}{49} + \frac{2 \sqrt{870}}{49} }} = 0 \]
It lands (y = 0) at t = 50/49 + 2*sqrt(870)/49.✓ Proved
\[ \left. 10 \sqrt{3} t \right|_{\substack{ t=\frac{50}{49} + \frac{2 \sqrt{870}}{49} }} = \frac{20 \sqrt{3} \left(25 + \sqrt{870}\right)}{49} \]
The horizontal distance then.✓ Proved

Answer \( H = \frac{348}{49} \approx 7.10,\ T = \frac{50}{49} + \frac{2 \sqrt{870}}{49} \approx 2.224,\ \text{range} = \frac{20 \sqrt{3} \left(25 + \sqrt{870}\right)}{49} \approx 38.53 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More projectile motion practice

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 20 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.672medium✓ Every equation proved

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.673medium✓ Every equation proved

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.674medium✓ Every equation proved

A projectile is fired from \( \displaystyle 2 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.675medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.676medium✓ Every equation proved

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.677medium✓ Every equation proved

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.679medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 40 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.680medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) ft above the ground at \( \displaystyle 96 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.681medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.682medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 64 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.683medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.684medium✓ Every equation proved

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 20 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.685medium✓ Every equation proved

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 96 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.752medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.753medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 40 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.754medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.755medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.756medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 64 \) ft/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.757medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 100 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.758medium✓ Every equation proved

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 20 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.759medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) ft above the ground at \( \displaystyle 64 \) ft/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.760medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.761medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 64 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.762medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 10 \) ft above the ground at \( \displaystyle 96 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.763medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 20 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.764medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.765medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.766medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 40 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.777medium✓ Every equation proved

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 64 \) ft/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.778medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.779medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 10 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.780medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.781medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 96 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.782medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 128 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.783medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.784medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) m above the ground at \( \displaystyle 100 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.785medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 40 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.786medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 128 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.819medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) m above the ground at \( \displaystyle 20 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.820medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 128 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.821medium✓ Every equation proved

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 128 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.822medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) ft above the ground at \( \displaystyle 128 \) ft/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.823medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 128 \) ft/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.824medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.825medium✓ Every equation proved

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.826medium✓ Every equation proved

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 80 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.827medium✓ Every equation proved

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 100 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.828medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 10 \) ft above the ground at \( \displaystyle 128 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.890medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.891medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 2 \) m above the ground at \( \displaystyle 100 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.892medium✓ Every equation proved

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 20 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.893medium✓ Every equation proved

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 100 \) m/s, \( \displaystyle 30^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.894medium✓ Nihil obstat

A projectile is fired from \( \displaystyle 0 \) m above the ground at \( \displaystyle 50 \) m/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.895medium✓ Every equation proved

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 64 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.896medium✓ Every equation proved

A projectile is fired from \( \displaystyle 10 \) m above the ground at \( \displaystyle 40 \) m/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 9.8 \) m/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.897medium✓ Every equation proved

A projectile is fired from \( \displaystyle 10 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 45^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.898medium✓ Every equation proved

A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.

Problem 9.899medium✓ Nihil obstat