Question

A ball is rolled off the edge of a horizontal table at a speed $4ms^{-1}$. It hits the ground after $0.4\;s$. Which of the following statements are true?
A. It hits the ground at a horizontal distance $1.6\;m$ from the edge of the table.
B. The speed with which it hits the ground is $4.0\;ms^{-1}$.
C. Height of the table is $0.8\;m$.
D. It hits off the ground at an angle of $60^{\circ}$ to the horizontal.

Answer 1

Hint: There are multiple correct options for this question, so eliminate the wrong statement by verification. Remember to resolve the velocity into horizontal and vertical components and use them to calculate horizontal and vertical (table height) displacements respectively. Also, account for gravity only in the vertical direction and keep in mind that the initial velocity is projected only in the horizontal direction. To this end, verify each of the options given above numerically until you arrive at the wrong statement.

Formula Used:
Kinematic equations of motion:
$S=ut+\dfrac{1}{2}gt^2$
$v=u+gt$

Complete answer:
We are given that the initial horizontal velocity of the ball is $u_x = 4\;ms^{-1}$ and the ball hits the floor after a time of $t=0.4\;s$. Therefore, the horizontal distance covered by the ball will be:
$s_{horizontal} = u_x t = 4 \times 0.4 = 1.6\;m$
The initial vertical velocity of the ball is $u_y=0$. The vertical velocity with which the ball hits the ground can be determined by using the kinematic equation of motion:
$v_y = u_y + g_y t = 0+(10 \times 0.4) = 4\;ms^{-1}$
The vertical distance covered by the ball is nothing but the height of the table from which it rolls off. The height of the table can also be calculated by using the kinematic equation of motion:
$ h_{table} = s_{vertical} = u_y t + \dfrac{1}{2}g_y t^2 = 0 + \dfrac{1}{2}(10 \times 0.4^2) = 0.8\;m$

If theta is the angle the ball makes with the vertical as it hits the ground,
$tan\theta = \dfrac{v_x}{v_y} = \dfrac{u_x}{v_y} = \dfrac{4}{-4} = -1 \Rightarrow \theta = tan^{-1}(-1) = -45^{\circ}$
Therefore, except for the last statement all other statements are true.
The correct statements would thus be :
A. It hits the ground at a horizontal distance $1.6\;m$ from the edge of the table.
B. The speed with which it hits the ground is $4.0\;ms^{-1}$.
C. Height of the table is $0.8\;m$.

Note:
Remember to resolve the velocity into its respective horizontal and vertical directions and use only those components of displacements and acceleration related to that direction at every point in the trajectory of the ball as it rolls off onto the ground. Do not forget that the acceleration due to gravity is only in the vertical direction as the change in this acceleration is negligible along the horizontal axis of motion of the ball.