Kinematics: displacement, velocity and acceleration Problem set A One question at a time

A particle moves in a straight line. The displacement from a fixed point is \(s\) m, where \(s=2 t^{3} - 9 t^{2} + 12 t + 1\) and \(t\ge0\) is in seconds.

1. (a) Find the velocity when \(t=3\).

   • \(v=\frac{ds}{dt}=6 t^{2} - 18 t + 12\)
   • \(v(3)=12\)

2. (b) Find the acceleration when \(t=3\).

   • \(a=\frac{dv}{dt}=12 t - 18\)
   • \(a(3)=18\)

A trolley moves along a straight track. The displacement from a fixed point is \(s=15e^{-0.3t}+t\) m, where \(t\ge0\) is in seconds. Give your answers to 3 significant figures.

1. (a) Find the velocity when \(t=2\).

   • \(v=1-4.5e^{-0.3t}\)
   • \(v(2)=−1.47\)

2. (b) Find the acceleration when \(t=2\).

   • \(a=1.35e^{-0.3t}\)
   • \(a(2)=0.741\)

A particle moves in a straight line. The displacement from a fixed point is \(s=2 t^{3} - 15 t^{2} + 24 t\) m, for \(t\ge0\) seconds.

1. (a) Find the times when the particle is at rest.

   • \(v=6 t^{2} - 30 t + 24\)
   • \(v=0\Rightarrow 6 \left(t - 4\right) \left(t - 1\right)=0\)
   • \(t=1,\ t=4\)

2. (b) Find the displacement at each of those times.

   • \(s(1)=11\)
   • \(s(4)=−16\)

A mass on a spring moves up and down. The displacement from its rest position is \(x=4\sin(3t)\) m, where \(t\) is in seconds. Use radians.

1. (a) Find the velocity when \(t=1\). Give your answer to 3 significant figures.

   • \(v=\frac{dx}{dt}=12 \cos{\left(3 t \right)}\)
   • \(v(1)=−11.9\)

2. (b) Find the first time after \(t=0\) when the object is at rest.

   • \(v=0\Rightarrow \cos(3t)=0\Rightarrow 3t=\frac{\pi}{2}\)
   • \(t=0.524\)

3. (c) Write down the greatest speed of the object.

   • \(|v|\le 12\)
   • The greatest speed is 12 m s⁻¹.

A particle moves in a straight line. Its displacement from O is \(s=t^{3} - 12 t^{2} + 36 t\) metres after \(t\) seconds, \(t\ge0\). For each time, say whether the particle is speeding up or slowing down.

1. (a) \(t=1\)

   • \(v=3 t^{2} - 24 t + 36,\ a=6 t - 24\)
   • \(v(1)=15,\ a(1)=−18\)
   • v and a have opposite signs, so the particle is slowing down.

2. (b) \(t=3\)

   • \(v=3 t^{2} - 24 t + 36,\ a=6 t - 24\)
   • \(v(3)=−9,\ a(3)=−6\)
   • v and a have the same sign, so the particle is speeding up.

3. (c) \(t=7\)

   • \(v=3 t^{2} - 24 t + 36,\ a=6 t - 24\)
   • \(v(7)=15,\ a(7)=18\)
   • v and a have the same sign, so the particle is speeding up.

A particle moves in a straight line. Its acceleration is \(a=6 t - 4\) m s\(^{-2}\). When \(t=0\) the velocity is 3 m s\(^{-1}\) and the displacement is 1 m.

1. (a) Find an expression for the velocity.

   • \(v=\int a\,dt=3 t^{2} - 4 t+c\)
   • \(v(0)=3\Rightarrow c=3\)

2. (b) Find an expression for the displacement.

   • \(s=\int v\,dt=t^{3} - 2 t^{2} + 3 t+k\)
   • \(s(0)=1\Rightarrow k=1\)

3. (c) Find the displacement when \(t=2\).

   • \(s(2)=7\)

A cyclist stops pedalling and freewheels along a straight road. Its distance from the start is \(s=60\left(1-e^{-0.2t}\right)\) metres after \(t\) seconds, \(t\ge0\).

1. (a) Find the velocity when \(t=5\). Give your answer to 3 significant figures.

   • \(v=12e^{-0.2t}\)
   • \(v(5)=4.41\)

2. (b) Find the time when the velocity is 1 m s\(^{-1}\). Give your answer to 3 significant figures.

   • \(12e^{-0.2t}=1\)
   • \(t=-\frac{1}{0.2}\ln\left(\frac{1}{12}\right)=12.4\)

3. (c) Find the distance that the cyclist approaches and never passes.

   • \(t\to\infty:\ e^{-0.2t}\to0,\ s\to60\)

A seat on a Ferris wheel moves round and up and down. Its height above the ground is \(h=11-9\cos(0.5t)\) m after \(t\) seconds, where \(t\ge0\). Use radians.

1. (a) Find the vertical velocity when \(t=3\). Give your answer to 3 significant figures.

   • \(v=\frac{dh}{dt}=4.5\sin(0.5t)\)
   • \(v(3)=4.49\)

2. (b) Find the greatest rate at which the seat rises or falls.

   • The sine function has greatest value 1.
   • The greatest rate is 4.5 m s⁻¹.

3. (c) Find the first time that the seat is rising at 3 m s\(^{-1}\). Give your answer to 3 significant figures.

   • \(4.5\sin(0.5t)=3\)
   • Solve the equation with the calculator (radian mode), or by taking the inverse sine.
   • The first time is t = 1.46 s.

A trolley moves along a straight track. Its velocity is \(v=t^{2} - 7 t + 10\) m s\(^{-1}\) for \(0\le t\le6\). Give your answers to 3 significant figures.

1. (a) Find the times when the velocity is zero.

   • \(t^{2} - 7 t + 10=0\)
   • \(t=2,\ t=5\)

2. (b) Find the displacement from the starting point after 6 seconds.

   • \(\int_{0}^{6}v\,dt=6\)
   • Forwards and backwards parts partly cancel.

3. (c) Find the total distance travelled in the 6 seconds.

   • \(\int_{0}^{6}|v|\,dt=15\)
   • The distance is 15 m.

A particle moves in a straight line. Its displacement from O is \(s=5te^{-0.4t}\) metres after \(t\) seconds, \(t\ge0\). Give your answers to 3 significant figures where needed.

1. (a) Find expressions for the velocity and the acceleration.

   • Product rule with \(u=5t\) and the exponential factor.
   • \(v=5(1-0.4t)e^{-0.4t}\)
   • \(a=2(0.4t-2)e^{-0.4t}\)

2. (b) Find the time when the particle is at rest, and its displacement then.

   • \(v=0\Rightarrow 1-0.4t=0\Rightarrow t=2.5\)
   • \(s(2.5)=4.60\)

3. (c) Find the time when the acceleration is zero, and the velocity then.

   • \(a=0\Rightarrow 0.4t-2=0\Rightarrow t=5\)
   • \(v(5)=−0.677\)
   • The velocity is negative: the particle is moving back towards O.

A particle moves in a straight line. Its velocity is \(v=5\sin(0.8t)-2\) m s\(^{-1}\) for \(0\le t\le8\). Use radians and give your answers to 3 significant figures.

1. (a) Find the times when the particle is at rest.

   • \(5\sin(0.8t)=2\)
   • \(t=0.514\text{ and }t=3.41\)
   • Use the calculator to find both solutions in the interval.

2. (b) Find the total distance travelled in the 8 seconds.

   • \(\int_{0}^{8}|v|\,dt=27.3\)
   • The velocity changes sign at both times, so use the absolute value.

3. (c) Find the displacement from the starting point after 8 seconds.

   • \(\int_{0}^{8}v\,dt=−16.0\)