The position of a crate sliding down a ramp is given by x=(0.25t^3) m, y=(1.5t^2) m, z=(6-0.75t^{\frac{5}{2}}) m, where t is in seconds. Determine the magnitude of the crate’s velocity and acceleration when t = 2 s.
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Solution:
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Remember that taking the first derivative of a position time equation gives velocity, and take the second derivative of the position time equation gives acceleration. Let us find the velocity and acceleration equations:
y=(1.5t^2) m
z=(6-0.75t^{\frac{5}{2}}) m
v_x=\dot{x}=0.75t^2 m/s
v_y=\dot{y}=3t m/s
v_z=\dot{z}=-1.875t^{\frac{3}{2}} m/s
a_x=\ddot{x}=1.5t m/s^2
a_y=\ddot{y}=3 m/s^2
a_z=\ddot{z}=-2.8125t^{\frac{1}{2}} m/s^2
Remember that a single dot on top represents the first derivative and two dots represent the second derivative. In our case, a single dot represents the velocity, and two dots represent acceleration.
v_x=0.75(2)^2=3 m/s
v_y=3(2)=6 m/s
v_z=-1.875(2)^{\frac{3}{2}}=-5.3 m/s
The magnitude of velocity is:
v=\sqrt{(3)^2+(6)^2+(-5.3)^2}=8.55 m/s
At t = 2 s, the acceleration is:
a_y=3 m/s^2
a_z=-2.8125(2)^{\frac{1}{2}}=-3.98 m/s^2
The magnitude of acceleration is:
a=\sqrt{(3)^2+(3)^2+(-3.98)^2}=5.82 m/s^2
Final Answers:
a=5.82 m/s^2