A projectile is fired with initial speed feet per second from a height of h feet at an angle of above the horizontal. Assuming that the only force acting on the object is gravity, find the speed at impact. Round to the nearest tenth of a foot.
A) 263.1 ft/s
B) 725.9 ft/s
C) 181.8 ft/s
D) 1022.4 ft/s

Question

A projectile is fired with initial speed  feet per second from a height of h feet at an angle of  above the horizontal. Assuming that the only force acting on the object is gravity, find the speed at impact. Round to the nearest tenth of a foot. 
A) 263.1 ft/s 
B) 725.9 ft/s 
C) 181.8 ft/s 
D) 1022.4 ft/s

Quizplus · Accepted Answer

We can use the following kinematic equations to solve for the final velocity at impact:

$\Delta y = v_{i}t + \frac{1}{2}at^2$

$v_{f}^2 = v_{i}^2 + 2a\Delta y$

Given that the projectile is fired from a height of h feet, and we are assuming that the only force acting on the object is gravity, we know that the acceleration is equal to -32 ft/s$^2$.

We can break the initial velocity into its x- and y-components as follows:

$v_{ix} = v_{i}\cos(11^\circ)$

$v_{iy} = v_{i}\sin(11^\circ)$

At the highest point of the trajectory, the y-velocity is equal to 0 (since the object is momentarily at rest before falling back down), so we can use the following kinematic equation to solve for the time it takes for the projectile to reach its maximum height:

$v_{iy} = v_{i}\sin(11^\circ) = at$

$t = \frac{v_{i}\sin(11^\circ)}{32}$

At impact, the projectile is back at its initial height of h feet, so we can use the following kinematic equation to solve for the final velocity:

$h = v_{iy}t + \frac{1}{2}at^2$

$h = v_{i}\sin(11^\circ)\left(\frac{v_{i}\sin(11^\circ)}{32}\right) + \frac{1}{2}(-32)\left(\frac{v_{i}\sin(11^\circ)}{32}\right)^2$

$h = \frac{v_{i}^2\sin^2(11^\circ)}{64} - \frac{v_{i}^2\sin^2(11^\circ)}{64}$

$h = 0$

This simplifies to:

$v_{f}^2 = v_{i}^2 + 2a\Delta y$

$v_{f}^2 = v_{i}^2 + 2(-32)(-h)$

$v_{f}^2 = v_{i}^2 + 64h$

Substituting the given values, we get:

$v_{f}^2 = (11eaa948_cdaa_4987_84bc_79dcce99b7c0_TB2342_11)^2 + 64(11eaa948_cdaa_4986_84bc_83b17603d33a_TB2342_11)$

$v_{f}^2 \approx 32767.1$

$v_{f} \approx 181.8$ ft/s

Therefore, the best choice is C.

A Projectile Is Fired with Initial Speed Feet Per