Solving for Vo: Ball Strikes Car Moving Along x-Axis

[JennV]

Homework Statement

A car that is initially at rest at the origin has an acceleration of 7.20 m/s2 along the positive x-axis (the ground). At the instant the car is at the origin, you mischeivously throw a ball H=34.9m directly above the car at 43.8 degrees below the horizontal at an unknown speed Vo in such a way as to eventually hit the car.

A: Neglecting air resistance, what initial speed is required for the ball to strike the car moving along the x-axis?

B: How long does it take for the ball to strike the car?

C: How fast is the ball moving at the instant it strikes the car?

Homework Equations

x(t)=Xo+Vo*cos(theta)t
Vx(t)=Vo*cos(theta)
y(t)=Vo*sin(theta)t-(1/2)gt^2
Vy(t)=Vo*sin(theta)-gt

The Attempt at a Solution

I tried plugging in values in these formulas, but I just got totally lost. All I'm hoping for is a starting point.

Thank you very much.

 

[anathema]

Hello,

To solve this problem, we will need to use the kinematic equations and the given information about the car and the ball to find the initial speed of the ball, the time it takes for the ball to hit the car, and the final speed of the ball at impact.

A: To find the initial speed of the ball, we can use the fact that the ball will eventually hit the car, which means that the x-coordinate of the ball at the time of impact will be the same as the x-coordinate of the car. We can set up an equation using the x-coordinate equation and solve for Vo:

x(t)=Xo+Vo*cos(theta)t
0 = 0 + Vo*cos(43.8)*t
Vo = 0/tan(43.8)
Vo = 0.85*t

B: To find the time it takes for the ball to hit the car, we can use the y-coordinate equation and set it equal to the height of the ball when it is thrown (H=34.9m). We can then solve for t:

y(t)=Vo*sin(theta)t-(1/2)gt^2
34.9 = Vo*sin(43.8)*t - (1/2)(9.8)t^2
34.9 = 0.85*t*tan(43.8)*t - 4.9t^2
34.9 = 0.85*t^2*tan(43.8) - 4.9t^2
0 = 0.85*t^2*tan(43.8) - 4.9t^2 - 34.9
Using the quadratic formula, we get t=3.12s or t=-5.61s. Since we are looking for the time it takes for the ball to hit the car, we can disregard the negative solution. Therefore, the time it takes for the ball to hit the car is approximately 3.12 seconds.

C: To find the final speed of the ball at impact, we can use the x-velocity equation (Vx(t)=Vo*cos(theta)) and plug in the time we found in part B to find the x-velocity at impact:

Vx(t)=Vo*cos(theta)
Vx(3.12)=0.85*cos(43.8)
Vx(3.12)=0.85*0.69
V